I am trying to learn about fluid dynamics in the context of incompressible, Newtonian fluids, Laminar flows. Unfortunately, different authors use different notations and every time I found my self very confused about some concepts.

In my knowledge, the total stress tensor: $$ \sigma_{ij} = -p\delta_{ij} + \tau_{ij}$$

Where $\delta_{ij}$ is the kronecker delta, and $p$ is the hydrostatic pressure given by: $$p = -{\sigma_{ii}\over 3}$$

My understanding is that the deviatoric tensor, $\tau_{ij}$ is called: the viscous stress tensor.

If so, what is the shear stress tensor? is it the same name for the viscous stress tensor?

I hope someone could help me clarify this ambiguity.

  • $\begingroup$ They are the same thing, deviatoric tensor, viscous tensor or shear stress tensor. They are different from zero if the 3 principal stresses $\sigma_{ii}$ are not all equal. $\endgroup$ May 3, 2020 at 1:33
  • $\begingroup$ @ClaudioSaspinski: Any reference? $\endgroup$ May 3, 2020 at 9:28
  • $\begingroup$ I think by shear stress tensor they mean the deviatoric stress tensor. $\endgroup$ May 3, 2020 at 20:20

1 Answer 1


In my textbook, Physical Hydrodynamics by Guyon, $\sigma_{ij}$ is called the stress tensor or I think wiki calls it Cauchy stress tensor. $\tau_{ij}$ here is called the viscous stress tensor or viscous shear stress tensor.

Liquid will flow under shear stress, so still liquid doesn't have shear stress, only stress that's perpendicular to any surface, which is $-p\delta_{ij}$. If we use still liquid as a reference point, the Cauchy stress tensor can be divided into this component and an extra component, which involves both the shear stress and the viscosity of the fluid, which is $\tau_{ij}$.

  • $\begingroup$ Could you explain this "Liquid will flow under shear stress, so still liquid doesn't have shear stress,"? what is "still liquid" in "if we use still liquid ..."? $\endgroup$ May 2, 2020 at 15:31
  • $\begingroup$ Sorry, your answer makes me more confused. $\endgroup$ May 2, 2020 at 15:34
  • $\begingroup$ Take a piece of paper and put it on top of the water surface. Use your hand to slide the paper forward and water will flow. Do the same thing on a solid and you would at most deform it into a Parallelepiped. One of the characteristics of liquid is that it doesn't sustain shear force, it flows when you apply shear force. $\endgroup$
    – Liuke LYU
    May 2, 2020 at 15:36
  • $\begingroup$ Sorry, And how is that related to my question? could you use equations to describe your ideas? I find your language hard for me to follow, if there are equation I will try my best to understand them. Thanks $\endgroup$ May 2, 2020 at 15:40
  • $\begingroup$ It's a motivation for us to use still water as a reference point and define viscous stress tensor as the total stress tensor minus this reference point. $\endgroup$
    – Liuke LYU
    May 2, 2020 at 15:42

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